The following problem is a generalization of a well known inequality:
Let be positive real numbers such that . Prove that
, where is a non-negative real number.
From Holder’s inequality we get that
, or .
So, it suffices to prove that
But the last relation is equivalent to .
Let us denote by the respectively. Then .
So, our inequality takes the form . This inequality is homogeneous so, we consider the sum equal to .
Doing some manipulations in left and right hand side we only need to prove that
Now from the AM-GM inequality we have that
The last one is equal to .
After that, we only need to prove
which is Schur’s 3rd degree inequality, Q.E.D.